<< Chapter < Page Chapter >> Page >

The half-life of a reaction is the time required to decrease the amount of a given reactant by one-half. The half-life of a zero-order reaction decreases as the initial concentration of the reactant in the reaction decreases. The half-life of a first-order reaction is independent of concentration, and the half-life of a second-order reaction decreases as the concentration increases.

Key equations

  • integrated rate law for zero-order reactions: [ A ] = k t + [ A ] 0 , t 1 / 2 = [ A ] 0 2 k
  • integrated rate law for first-order reactions: ln [ A ] = k t + ln [ A ] 0 , t 1 / 2 = 0.693 k
  • integrated rate law for second-order reactions: 1 [ A ] = k t + 1 [ A ] 0 , t 1 / 2 = 1 [ A ] 0 k

Chemistry end of chapter exercises

Describe how graphical methods can be used to determine the order of a reaction and its rate constant from a series of data that includes the concentration of A at varying times.

Got questions? Get instant answers now!

Use the data provided to graphically determine the order and rate constant of the following reaction: SO 2 Cl 2 SO 2 + Cl 2

Time (s) 0 5.00 × 10 3 1.00 × 10 4 1.50 × 10 4
[SO 2 Cl 2 ] ( M ) 0.100 0.0896 0.0802 0.0719
Time (s) 2.50 × 10 4 3.00 × 10 4 4.00 × 10 4
[SO 2 Cl 2 ] ( M ) 0.0577 0.0517 0.0415

Plotting a graph of ln[SO 2 Cl 2 ] versus t reveals a linear trend; therefore we know this is a first-order reaction:
A graph is shown with the label “Time ( s )” on the x-axis and “l n [ S O subscript 2 C l subscript 2 ] M” on the y-axis. The x-axis begins at 0 and extends to 4.00 times 10 superscript 4 with markings every 1.00 times 10 superscript 4. The y-axis shows markings extending from negative 3.5 to negative 2.5. A decreasing linear trend line is drawn through seven points at the approximate coordinates: (0, negative 2.3), (0.5 times 10 superscript 4, negative 2.4), (1.0 times 10 superscript 4, negative 2.5), (1.5 times 10 superscript 4, negative 2.6), (2.0 times 10 superscript 4, negative 2.9), (2.5 times 10 superscript 4, negative 3.0), and (3.0 times 10 superscript 4, negative 3.2).
k = −2.20 × 10 5 s −1

Got questions? Get instant answers now!

Use the data provided in a graphical method to determine the order and rate constant of the following reaction:
2 P Q + W

Time (s) 9.0 13.0 18.0 22.0 25.0
[P] ( M ) 1.077 × 10 −3 1.068 × 10 −3 1.055 × 10 −3 1.046 × 10 −3 1.039 × 10 −3

Got questions? Get instant answers now!

Pure ozone decomposes slowly to oxygen, 2O 3 ( g ) 3O 2 ( g ) . Use the data provided in a graphical method and determine the order and rate constant of the reaction.

Time (h) 0 2.0 × 10 3 7.6 × 10 3 1.00 × 10 4
[O 3 ] ( M ) 1.00 × 10 −5 4.98 × 10 −6 2.07 × 10 −6 1.66 × 10 −6
Time (h) 1.23 × 10 4 1.43 × 10 4 1.70 × 10 4
[O 3 ] ( M ) 1.39 × 10 −6 1.22 × 10 −6 1.05 × 10 −6


A graph is shown with the label, “t ( h ,)” on the x-axis and, “1 divided by [ O subscript 3 ] M,” on the y-axis. The x-axis shows markings at 0, 2 times 10 superscript 3, 6 times 10 superscript 3, 10 time 10 superscript 3, 14 times 10 superscript 3, and 18 times 10 superscript 3. The y-axis shows markings beginning at 0, increasing by 1 up to and including 9. An increasing linear trend line is drawn through seven points at the coordinates: (0, 1.00), (2.0 times 10 superscript 3, 2.01), (7.6 times 10 superscript 3, 4.83), (1.00 times 10 superscript 4, 6.02), (1.23 times 10 superscript 4 , 6.02), (1.43 times 10 superscript 4, 8.20) and (1.70 times 10 superscript 4, 9.52). A horizontal line segment is drawn through the first point and a vertical line segment is similarly drawn through the last point to make a right triangle on the graph. The horizontal leg of the triangle is labeled “ capital delta t.” The vertical leg is labeled “capital delta 1 divided by [ O subscript 3 ].”
The plot is nicely linear, so the reaction is second order.
k = 50.1 L mol −1 h −1

Got questions? Get instant answers now!

From the given data, use a graphical method to determine the order and rate constant of the following reaction:
2 X Y + Z

Time (s) 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
[ X ] ( M ) 0.0990 0.0497 0.0332 0.0249 0.0200 0.0166 0.0143 0.0125

Got questions? Get instant answers now!

What is the half-life for the first-order decay of phosphorus-32? ( 15 32 P 16 32 S + e ) The rate constant for the decay is 4.85 × 10 −2 day −1 .

14.3 d

Got questions? Get instant answers now!

What is the half-life for the first-order decay of carbon-14? ( 6 14 C 7 14 N + e ) The rate constant for the decay is 1.21 × 10 −4 year −1 .

Got questions? Get instant answers now!

What is the half-life for the decomposition of NOCl when the concentration of NOCl is 0.15 M ? The rate constant for this second-order reaction is 8.0 × 10 −8 L/mol/s.

8.3 × 10 7 s

Got questions? Get instant answers now!

What is the half-life for the decomposition of O 3 when the concentration of O 3 is 2.35 × 10 −6 M ? The rate constant for this second-order reaction is 50.4 L/mol/h.

Got questions? Get instant answers now!

The reaction of compound   A to give compounds   C and   D was found to be second-order in   A . The rate constant for the reaction was determined to be 2.42 L/mol/s. If the initial concentration is 0.500 mol/L, what is the value of t 1/2 ?

0.826 s

Got questions? Get instant answers now!

The half-life of a reaction of compound A to give compounds D and E is 8.50 min when the initial concentration of A is 0.150 mol/L. How long will it take for the concentration to drop to 0.0300 mol/L if the reaction is (a) first order with respect to A or (b) second order with respect to A ?

Got questions? Get instant answers now!

Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of 3 × 10 4 g/mol that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate equation that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg (0.15 × 10 −6 g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.

[Penicillin] ( M ) Rate (mol/L/min)
2.0 × 10 −6 1.0 × 10 −10
3.0 × 10 −6 1.5 × 10 −10
4.0 × 10 −6 2.0 × 10 −10

The reaction is first order.
k = 1.0 × 10 7 mol −1 min −1

Got questions? Get instant answers now!

Both technetium-99 and thallium-201 are used to image heart muscle in patients with suspected heart problems. The half-lives are 6 h and 73 h, respectively. What percent of the radioactivity would remain for each of the isotopes after 2 days (48 h)?

Got questions? Get instant answers now!

There are two molecules with the formula C 3 H 6 . Propene, CH 3 CH = CH 2 , is the monomer of the polymer polypropylene, which is used for indoor-outdoor carpets. Cyclopropane is used as an anesthetic:
A structural formula for cyclopropane is shown. Three C H subscript 2 groups are positioned as vertices of an equilateral triangle connected with single bonds represented by line segments.

When heated to 499 °C, cyclopropane rearranges (isomerizes) and forms propene with a rate constant of
5.95 × 10 −4 s −1 . What is the half-life of this reaction? What fraction of the cyclopropane remains after 0.75 h at 499.5 °C?

4.98; 20% remains

Got questions? Get instant answers now!

Fluorine-18 is a radioactive isotope that decays by positron emission to form oxygen-18 with a half-life of 109.7 min. (A positron is a particle with the mass of an electron and a single unit of positive charge; the equation is 518 9 F 18 8 O + e . ) Physicians use 18 F to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient. The glucose accumulates in the regions where the brain is active and needs nourishment.

(a) What is the rate constant for the decomposition of fluorine-18?

(b) If a sample of glucose containing radioactive fluorine-18 is injected into the blood, what percent of the radioactivity will remain after 5.59 h?

(c) How long does it take for 99.99% of the 18 F to decay?

Got questions? Get instant answers now!

Suppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for 1 64 of the initial dose to remain in the athlete’s body?

252 days

Got questions? Get instant answers now!

Recently, the skeleton of King Richard III was found under a parking lot in England. If tissue samples from the skeleton contain about 93.79% of the carbon-14 expected in living tissue, what year did King Richard III die? The half-life for carbon-14 is 5730 years.

Got questions? Get instant answers now!

Nitroglycerine is an extremely sensitive explosive. In a series of carefully controlled experiments, samples of the explosive were heated to 160 °C and their first-order decomposition studied. Determine the average rate constants for each experiment using the following data:

Initial [C 3 H 5 N 3 O 9 ] ( M ) 4.88 3.52 2.29 1.81 5.33 4.05 2.95 1.72
t (s) 300 300 300 300 180 180 180 180
% Decomposed 52.0 52.9 53.2 53.9 34.6 35.9 36.0 35.4

[ A ] 0 ( M ) k × 10 3 (s −1 )
4.88 2.45
3.52 2.51
2.29 2.54
1.81 2.58
5.33 2.35
4.05 2.44
2.95 2.47
1.72 2.43

Got questions? Get instant answers now!

For the past 10 years, the unsaturated hydrocarbon 1,3-butadiene ( CH 2 = CH CH = CH 2 ) has ranked 38th among the top 50 industrial chemicals. It is used primarily for the manufacture of synthetic rubber. An isomer exists also as cyclobutene:
A structural formula for cyclobutene is shown. The figure has two C H subscript 2 groups as the upper two vertices of a square structure. These groups are connected by a single, short line segment. Line segments extend below each of these C H subscript 2 groups to C H groups positioned at the lower two vertices of the square structure. The C H groups are connected with a double line segment indicating a double bond.

The isomerization of cyclobutene to butadiene is first-order and the rate constant has been measured as 2.0 × 10 −4 s −1 at 150 °C in a 0.53-L flask. Determine the partial pressure of cyclobutene and its concentration after 30.0 minutes if an isomerization reaction is carried out at 150 °C with an initial pressure of 55 torr.

Got questions? Get instant answers now!

Questions & Answers

An atom or group of atoms bearing anelectrical charge such as the sodium and chlorine atoms in a salt solution.
Adazion Reply
Hello guys! Answer me questions nah
Adazion
pls wat is periodic table
Prince
it's a list that shows the chemical element arranged according to their properties.
Adazion Reply
what is the chemical equation for ideal gas?
Adazion
what's Boyle and gas law?
Adazion
what's the meaning of this℃ in atomic table
Adazion
wat are ions
Sinyene
What is periodic table
SIRAJO Reply
Table that shows the elements in order of their atomic number
Uzair
organization of everything known in the universe in groups and periods. The structure is based on increasing mass and reaction properties.
Kate
How to mix chemical
Ukeh Reply
why the elements of group 7 are called Noble gases
isaac Reply
they aren't. group 8 is the noble gasses. they are snobs that don't mix with others like nobles, they have full valence shells so they don't form bonds with other elements easily. nobles don't mingle with the common folk...
Jessica
the group 7elements are not the noble gases . according to modern periodic group 18 are called noble gases elements because their valence shell are completely field so that they can't gain or loss electron so they are not able to involve in any chemical reaction.
Leena
Group 7 element they are not noble gases they halogen and halogen mean salt formers
SIRAJO
what is chemistry
Daniel Reply
chemistry is the branch of science which deal with the composition of matter
SHEDRACK
chemistry is an organized way to think about matter
Kate
Chemistry is the branch of science that deals with the composition, properties and uses of matter
Bamgbose
The branch of science concerned with the substances of which matter is composed, the investigation of their properties and reactions, and the use of such reactions to form new substances.
Aminu
discuss the orbital stracture of the following methane,ethane,ethylene,acetylene
khadija Reply
Why phosphurs in solid state have one atom but in gas state have four atoms
Shehab Reply
Are nuclear reactions both exothermic reactions and endothermic reactions or what?
Blessed Reply
to what volume must 8.32 NaOH be diluted to its analytical concentration 0.20 M
Sheriza Reply
weight in mg 1.76 mole of I
Sheriza
the types of hydrocarbons
Ohanaka Reply
u are mad go and open textbook
Emmanuel
hahahahahahahahahahahahaha
Jessica
aliphatic and aromatic hydrocarbons
Osakue
stupid boy Emmanuel
Ohanaka
saturated and unsaturated
Leena
aromatic hydrocarbon aliphatic hydrocarbon
SIRAJO
I don't use to see the messages
Adazion Reply
Hhhhhh
SIRAJO
how can you determine the electronegativity of a compound or in molecules
Shalom Reply
when u move from left to right in a periodic table the negativity increases
reeza
Are you trying to say that the elctronegativity increases down the group and decreases across the period?
Ohanaka
yes and also increases across the period
reeza
for instance when you look at one group of elements in a periodic table electronegativity decreases when you go across the table electronegativity increases. hydrogen is more electronegative than sodium, potassium of that group. oxygen is more electronegative than carbon.
reeza
i hope we all know that organic compounds have carbon as their back bone
Madueke
OK,Thank you so much for the answer. I am happy now
Adazion Reply
can I ask you a question now
Osakue
yes
hanna
what is the oxidation number of nitrogen, oxygen and sulphur
Osakue
5, -2 & -2
hanna
What is periodic table
SIRAJO
What is an atom?
Adazion Reply
is a smallest particle of a chemical element that can exist
Osakue
can I ask a question
Osakue
it is a substance that cannot be broken down into simpler units by any chemical reaction
Madueke
An atom is the smallest part of an element dat can take part in chemical reaction.
Idris
an atom is the smallest part of an element that can take part in a chemical reaction nd still retain it chemical properties
Precious
Is the smallest particles of an element that take part in chemical reaction without been change
John
Practice Key Terms 2

Get the best Chemistry course in your pocket!





Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Chemistry' conversation and receive update notifications?

Ask